locusssssss

Case I..if the angle b/w the tangents is 90°..then the locus of that point will be x² + y² = 2a² ..if the equation of the circle is x² + y² = a²

Case II : If the angle b/w the tangents is 2θ ..
take the centre of the circle at origin and radius is r..
then the distance of the point from where the tangent is drawn will be r/sinθ ..by trigonometry..
so the locus will be x² + y² = r² / sin²Î¸..

and if it is not at origin???

If the centre is not at origin but at some general point (h,k) ..then u need to shift the origin and then the eqⁿ will be
(x-h)² + (y-k)² = r² / sin²Î¸